Sums of singular series with large sets and the tail of the distribution of primes

In 1976, Gallagher showed that the Hardy–Littlewood conjectures on prime k-tuples imply that the distribution of primes in log-size intervals is Poissonian. He did so by computing average values of the singular series constants over different sets of a fixed size k contained in an interval [1,h] as h → ∞, and then using this average to compute moments of the distribution of primes. In this paper, we study averages where k is relatively large with respect to h. We then apply these averages to the tail of the distribution. For example, we show, assuming appropriate Hardy–Littlewood conjectures and in certain ranges of the parameters, the number of intervals [n,n + λlogx] with n ≤ x containing at least k primes is ≪ x exp(−k/(λe)).